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What is m^3 – 3m^2 = 5m – 15 factored?

It's interesting that the problem was posed this way because you might notice that you can use the Distributive Property on both sides:

m^2(m – 3) = 5(m – 3)

At this point you might decide to say "If m ≠ 3 then I can divide both sides by (m–3) to get m^2 = 5, where m = ±√(5) are the only two answers."

However, if instead you subtracted 5(m – 3) from both sides:

m^2(m – 3) – 5(m – 3) = 0

and used the Distributive Property:

(m – 3)(m^2 – 5) = 0

Substitute (√(5))^2 for 5:

(m – 3)(m^2 – (√(5))^2) = 0

Use Difference of Squares:

(m – 3)(m + √(5))(m – √(5)) = 0

That's the answer to, "What is m^3 – 3m^2 = 5m – 15 factored?"

But now, using the Zero Product Property, you can find the values of m that make it true:

(m – 3) = 0 ==> m = 3

(m – √(5)) = 0 ==> m = + √(5)

(m + √(5)) = 0 ==> m = – √(5)

Hi Tabalina;

m^3-3m^2=5m-15

Let's bring everything to one side...

m^{3}-3m^{2}-5m+15=0

For the FOIL...

FIRST must be (m^{2})(m)=m^{3}

(m^{2}...)(m...)

OUTER must be (-3)(m^{2})=-3m^{2}

(m^{2}...)(m-3)

INNER must be (-5)(m)=-5m

(m^{2}-5)(m-3)

LAST must be (-5)(-3)=15

0=(m^{2}-5)(m-3)

Either or both parenthetical equation(s) must equal zero...

0=m^{2}-5.........0=m-3

5=m^{2}............3=m

+/- √5=m

Please remember that - √5, when squared, produces the same result as +√5, when squared.

There are three answers...

m=-√5, +√5 and 3

Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...

The first step is to reorganize the equation to get a zero on the right hand side

m^{3} - 3m^{2} - 5 m + 15 = 0

The cubic expression on the left hand side can be factored by a method called factoring by grouping.

This method does not always work, but is very good when it does.

Step 1: pull a factor of m^{2} out of the first two terms m^{2} (m -3)

Step 2: pull a factor of -5 out of the last two terms -5(m-3)

Step 3: {this is the good part} notice that in both steps 1 and 2 we got a factor of (m-3)

Step 4: take advantage of this good fortune to write the whole expression as (m-3) ( m^{2} -5)

This ends the factor by grouping part

Step 5: m^{2} -5 factors as (m- sqrt(5) ) (m + sqrt(5))

Final answer (m-3) (m - sqrt(5)) ( m+ sqrt(5)) = 0

So the solutions to the problem are

m = 3

m = - sqrt(5)

m = + sqrt(5)

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